Calculate PMCC from raw data

9 It is thought that the pH value of sand (a measure of the sand's acidity) may affect the extent to which a particular species of plant will grow in that sand. A botanist wished to determine whether there was any correlation between the pH value of the sand on certain sand dunes, and the amount of each of two plant species growing there. She chose random sections of equal area on each of eight sand dunes and measured the pH values. She then measured the area within each section that was covered by each of the two species. The results were as follows. The results for species \(P\) can be summarised by $$n = 8 , \quad \Sigma x = 66.5 , \quad \Sigma x ^ { 2 } = 558.75 , \quad \Sigma y = 1935 , \quad \Sigma y ^ { 2 } = 711275 , \quad \Sigma x y = 17082.5 .$$

1. Give a reason why it might be appropriate to calculate the equation of the regression line of \(y\) on \(x\) rather than \(x\) on \(y\) in this situation.
2. Calculate the equation of the regression line of \(y\) on \(x\) for species \(P\), in the form \(y = a + b x\), giving the values of \(a\) and \(b\) correct to 3 significant figures.
3. Estimate the value of \(y\) for species \(P\) on sand where the pH value is 7.0 . The values of the product moment correlation coefficient between \(x\) and \(y\) for species \(P\) and \(Q\) are \(r _ { P } = 0.828\) and \(r _ { Q } = 0.0302\).
4. Describe the relationship between the area covered by species \(Q\) and the pH value.
5. State, with a reason, whether the regression line of \(y\) on \(x\) for species \(P\) will provide a reliable estimate of the value of \(y\) when the pH value is
   1. 8,
   2. 4 .
   3. Assume that the equation of the regression line of \(y\) on \(x\) for species \(Q\) is also known. State, with a reason, whether this line will provide a reliable estimate of the value of \(y\) when the pH value is 8 .

1. A French test and a Spanish test were sat by 11 students.

The table below shows their marks. Greg says that if these points were plotted on a scatter diagram, then the point \(( 30,90 )\) would be an outlier because 90 is an outlier for the Spanish marks. An outlier is defined as a value that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or smaller than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$

1. Show that 90 is an outlier for the Spanish marks. Ignoring the point (30, 90), Greg calculated the following summary statistics. $$\sum f = 422 \quad \sum s = 382 \quad S _ { f f } = 1667.6 \quad S _ { f s } = 1735.6$$
2. Use these summary statistics to show that the equation of the least squares regression line of \(s\) on \(f\) for the remaining 10 students is $$s = - 5.72 + 1.04 f$$ where the values of the intercept and gradient are given to 3 significant figures. You must show your working.
3. Give an interpretation of the gradient of the regression line. Two further students sat the French test but missed the Spanish test.
4. Using the equation given in part (b), estimate
   1. a Spanish mark for the student who scored 55 marks in their French test,
   2. a Spanish mark for the student who scored 18 marks in their French test.
5. State, giving a reason, which of the two estimates found in part (d) would be the more reliable estimate.

1 A geography student chose a certain point in a stream and took measurements of the speed of flow, \(v \mathrm {~ms} ^ { - 1 }\), of water at various depths, \(d \mathrm {~m}\), below the surface at that point. The results are shown in the table. \(n = 9 \quad \sum d = 2.7 \quad \sum v = 9.0 \quad \sum d ^ { 2 } = 0.96 \quad \sum v ^ { 2 } = 10.4 \quad \sum \mathrm {~d} v = 2.85\)

1. 1. Explain why \(d\) is an example of an independent, controlled variable.
   2. Use two relevant terms to describe the variable \(v\) in a similar way. A statistician believes that the point ( \(0.5,0.4\) ) may be an anomaly.
2. Calculate the equation of the least squares regression line of \(v\) on \(d\) for all the points in the table apart from ( \(0.5,0.4\) ).
3. Use the equation of the line found in part (b) to estimate the value of \(v\) when \(d = 0.5\).
4. Use your answer to part (c) to comment on the statistician's belief.
5. Use the diagram in the Printed Answer Booklet (which does not illustrate the data in this question) to explain what is meant by "least squares regression line".

3 An investor obtains data about the profits of 8 randomly chosen investment accounts over two one-year periods. The profit in the first year for each account is \(p \%\) and the profit in the second year for each account is \(q \%\). The results are shown in the table and in the scatter diagram. \(n = 8 \quad \sum \mathrm { p } = 28.5 \quad \sum \mathrm { q } = 26.7 \quad \sum \mathrm { p } ^ { 2 } = 136.35 \quad \sum \mathrm { q } ^ { 2 } = 116.35 \quad \sum \mathrm { pq } = 116.70\) \includegraphics[max width=\textwidth, alt={}, center]{bf1468d1-e02e-47d2-bf41-5bc8f5b4d7c4-3_782_1280_998_242}

1. State which, if either, of the variables \(p\) and \(q\) is independent.
2. Calculate the equation of the regression line of \(q\) on \(p\).
3. 1. Use the regression line to estimate the value of \(q\) for an investment account for which \(p = 2.5\).
   2. Give two reasons why this estimate could be considered reliable.
4. Comment on the reliability of using the regression line to predict the value of \(q\) when \(p = 7.0\).

1 Five observations of bivariate data \(( x , y )\) are given in the table.

1. Find the value of Pearson's product-moment correlation coefficient.
2. State what your answer to part (a) tells you about a scatter diagram representing the data.
3. A new variable \(a\) is defined by \(a = 3 x + 4\). Dee says "The value of Pearson's product-moment correlation coefficient between \(a\) and \(y\) will not be the same as the answer to part (a)." State with a reason whether you agree with Dee. An investor obtains data about the profits of 8 randomly chosen investment accounts over two one-year periods. The profit in the first year for each account is \(p \%\) and the profit in the second year for each account is \(q \%\). The results are shown in the table and in the scatter diagram.

   Account	A	B	C	D	E	F	G	H
   \(p\)	1.6	2.1	2.4	2.7	2.8	3.3	5.2	8.4
   \(q\)	1.6	2.3	2.2	2.2	3.1	2.9	7.6	4.8

   \(n = 8 \quad \Sigma p = 28.5 \quad \Sigma q = 26.7 \quad \Sigma p ^ { 2 } = 136.35 \quad \Sigma q ^ { 2 } = 116.35 \quad \Sigma p q = 116.70\) \includegraphics[max width=\textwidth, alt={}, center]{4c7546b9-03ee-47a1-915f-41e2b4ca19c0-03_762_1248_906_260}
   1. State which, if either, of the variables \(p\) and \(q\) is independent.
   2. Calculate the equation of the regression line of \(q\) on \(p\).
   3. 1. Use the regression line to estimate the value of \(q\) for an investment account for which \(p = 2.5\).
      2. Give two reasons why this estimate could be considered reliable.
   4. Comment on the reliability of using the regression line to predict the value of \(q\) when \(p = 7.0\).